Analysis of Algorithms Input Algorithm Analysis of Algorithms

Analysis of Algorithms Input Algorithm Analysis of Algorithms Output

Outline and Reading Running time (§ 3. 1) Pseudo-code (§ 3. 2) Counting primitive operations (§ 3. 3 -3. 5) Asymptotic notation (§ 3. 6) Asymptotic analysis (§ 3. 7) Case study Analysis of Algorithms 2

Running Time The running time of an algorithm varies with the input and typically grows with the input size Average case difficult to determine We focus on the worst case running time n n Easier to analyze Crucial to applications such as games, finance and robotics Analysis of Algorithms 3

Experimental Studies Write a program implementing the algorithm Run the program with inputs of varying size and composition Use a method like System. current. Time. Millis () to get an accurate measure of the actual running time Plot the results Analysis of Algorithms 4

Limitations of Experiments It is necessary to implement the algorithm, which may be difficult Results may not be indicative of the running time on other inputs not included in the experiment. In order to compare two algorithms, the same hardware and software environments must be used Analysis of Algorithms 5

Theoretical Analysis Uses a high-level description of the algorithm instead of an implementation Takes into account all possible inputs Allows us to evaluate the speed of an algorithm independent of the hardware/software environment Analysis of Algorithms 6

Pseudocode Example: find max High-level description element of an array of an algorithm More structured than Algorithm array. Max(A, n) English prose Input array A of n integers Less detailed than a Output maximum element of A program current. Max A[0] Preferred notation for i 1 to n 1 do describing algorithms if A[i] current. Max then Hides program design current. Max A[i] issues return current. Max Analysis of Algorithms 7

Pseudocode Details Method call Control flow n n n if … then … [else …] while … do … repeat … until … for … do … Indentation replaces braces Method declaration Algorithm method (arg [, arg…]) Input … Output … var. method (arg [, arg…]) Return value return expression Expressions Assignment (like in Java) Equality testing (like in Java) n 2 Superscripts and other mathematical formatting allowed Analysis of Algorithms 8

Primitive Operations Basic computations performed by an algorithm Identifiable in pseudocode Largely independent from the programming language Exact definition not important (we will see why later) Analysis of Algorithms Examples: n n n Evaluating an expression Assigning a value to a variable Indexing into an array Calling a method Returning from a method 9

Counting Primitive Operations By inspecting the pseudocode, we can determine the maximum number of primitive operations executed by an algorithm, as a function of the input size Algorithm array. Max(A, n) current. Max A[0] for i 1 to n 1 do if A[i] current. Max then current. Max A[i] { increment counter i } return current. Max # operations 2 2+n 2(n 1) 1 Total Analysis of Algorithms 7 n 1 10

Estimating Running Time Algorithm array. Max executes 7 n 1 primitive operations in the worst case Define a Time taken by the fastest primitive operation b Time taken by the slowest primitive operation Let T(n) be the actual worst-case running time of array. Max. We have a (7 n 1) T(n) b(7 n 1) Hence, the running time T(n) is bounded by two linear functions Analysis of Algorithms 11

Growth Rate of Running Time Changing the hardware/ software environment n n Affects T(n) by a constant factor, but Does not alter the growth rate of T(n) The linear growth rate of the running time T(n) is an intrinsic property of algorithm array. Max Analysis of Algorithms 12

Growth Rates Growth rates of functions: n n n Linear n Quadratic n 2 Cubic n 3 In a log-log chart, the slope of the line corresponds to the growth rate of the function Analysis of Algorithms 13

Constant Factors The growth rate is not affected by n n constant factors or lower-order terms Examples n n 102 n + 105 is a linear function 105 n 2 + 108 n is a quadratic function Analysis of Algorithms 14

Big-Oh Notation Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constants c and n 0 such that f(n) cg(n) for n n 0 Example: 2 n + 10 is O(n) n n 2 n + 10 cn (c 2) n 10/(c 2) Pick c 3 and n 0 10 Analysis of Algorithms 15

Big-Oh Notation (cont. ) Example: the function n 2 is not O(n) n n 2 cn n c The above inequality cannot be satisfied since c must be a constant Analysis of Algorithms 16

Big-Oh and Growth Rate The big-Oh notation gives an upper bound on the growth rate of a function The statement “f(n) is O(g(n))” means that the growth rate of f(n) is no more than the growth rate of g(n) We can use the big-Oh notation to rank functions according to their growth rate g(n) grows more f(n) grows more Same growth f(n) is O(g(n)) g(n) is O(f(n)) Yes No Yes Analysis of Algorithms 17

Classes of Functions Let {g(n)} denote the class (set) of functions that are O(g(n)) We have {n} {n 2} {n 3} {n 4} {n 5} … where the containment is strict {n 3} {n 2} {n} Analysis of Algorithms 18

Big-Oh Rules If is f(n) a polynomial of degree d, then f(n) is O(nd), i. e. , 1. 2. Drop lower-order terms Drop constant factors Use the smallest possible class of functions n Say “ 2 n is O(n)” instead of “ 2 n is O(n 2)” Use the simplest expression of the class n Say “ 3 n + 5 is O(n)” instead of “ 3 n + 5 is O(3 n)” Analysis of Algorithms 19

Asymptotic Algorithm Analysis The asymptotic analysis of an algorithm determines the running time in big-Oh notation To perform the asymptotic analysis n n We find the worst-case number of primitive operations executed as a function of the input size We express this function with big-Oh notation Example: n n We determine that algorithm array. Max executes at most 7 n 1 primitive operations We say that algorithm array. Max “runs in O(n) time” Since constant factors and lower-order terms are eventually dropped anyhow, we can disregard them when counting primitive operations Analysis of Algorithms 20

Computing Prefix Averages We further illustrate asymptotic analysis with two algorithms for prefix averages The i-th prefix average of an array X is average of the first (i + 1) elements of X A[i] (X[0] + X[1] + … + X[i])/(i+1) Computing the array A of prefix averages of another array X has applications to financial analysis Analysis of Algorithms 21

Prefix Averages (Quadratic) The following algorithm computes prefix averages in quadratic time by applying the definition Algorithm prefix. Averages 1(X, n) Input array X of n integers Output array A of prefix averages of X #operations A new array of n integers n for i 0 to n 1 do n s X[0] n for j 1 to i do 1 + 2 + …+ (n 1) s s + X[j] 1 + 2 + …+ (n 1) A[i] s / (i + 1) n return A 1 Analysis of Algorithms 22

Arithmetic Progression The running time of prefix. Averages 1 is O(1 + 2 + …+ n) The sum of the first n integers is n(n + 1) / 2 n There is a simple visual proof of this fact Thus, algorithm prefix. Averages 1 runs in O(n 2) time Analysis of Algorithms 23

Prefix Averages (Linear) The following algorithm computes prefix averages in linear time by keeping a running sum Algorithm prefix. Averages 2(X, n) Input array X of n integers Output array A of prefix averages of X A new array of n integers s 0 for i 0 to n 1 do s s + X[i] A[i] s / (i + 1) return A #operations n 1 n n n 1 Algorithm prefix. Averages 2 runs in O(n) time Analysis of Algorithms 24

Asymptotic Notation (terminology): Special classes of algorithms: logarithmic: linear: quadratic: O(n 2) polynomial: exponential: O(log n) O(nk), k ≥ 1 O(an), n > 1 “Relatives” of the Big-Oh n n (f(n)): Big Omega--asymptotic lower bound (f(n)): Big Theta--asymptotic tight bound Analysis of Algorithms 25

A table of functions wrt input n, assume that each primitive operation takes one microsecond (1 second = 106 microsecond). O(g(n)) 1 Second 1 Hour log 2 n 10300000 1010 n 1012 1. 3*1019 6. 8*1024 9. 7*1030 n 106 3. 6*109 2. 6*1012 3. 12*1015 n log 2 n 105 109 1011 1014 n 2 1000 6*104 1. 6*106 5. 6*107 n 3 100 1500 14000 1500000 2 n 19 31 41 51 n! 9 12 15 17 9 1 Month 100. 8*10 Analysis of Algorithms 1 Century 12 1010 15 26